I show that intuitive and logical considerations do not justify introducing Leibniz’s Law of the Indiscernibility of Identicals in more than a limited form, as applying to atomic formulas. Once this is accepted, it follows that Leibniz’s Law generalises to all formulas of the first-order Predicate Calculus but not to modal formulas. Among other things, identity turns out to be logically contingent.

Composition as Identity is the view that an object is identical to its parts taken collectively. I elaborate and defend a theory based on this idea: composition is a kind of identity. Since this claim is best presented within a plural logic, I develop a formal system of plural logic. The principles of this system differ from the standard views on plural logic because one of my central claims is that identity is a relation which comes in a variety of (...) forms and only one of them obeys substitution unrestrictedly. I justify this departure from orthodoxy by showing some problems which result from attempts to avoid inconsistencies within plural logic by means of postulating other non-singular terms besides plural terms. Thereby, some of the main criticisms raised against Composition as Identity can be addressed. Further, I argue that the way objects are arranged is relevant with respect to the question which object they compose, i.e. to which object they are identical to. This helps to meet a second group of arguments against Composition as Identity. These arguments aim to show that identifying composite objects on the basis of the identity of their parts entails, contrary to our common sense view, that rearranging the parts of a composite object does not leave us with a different object. Moreover, it allows us to carve out the intensional aspects of Composition as Identity and to defend mereological universalism, the claim that any objects compose some object. Much of the pressure put on the latter view can be avoided by distinguishing the question whether some objects compose an object from the question what object they compose. Eventually, I conclude that Composition as Identity is a coherent and plausible position, as long as we take identity to be a more complex relation than commonly assumed. (shrink)

It is widely taken that the first-order part of Frege's Begriffsschrift is complete. However, there does not seem to have been a formal verification of this received claim. The general concern is that Frege's system is one axiom short in the first-order predicate calculus comparing to, by now, the standard first-order theory. Yet Frege has one extra inference rule in his system. Then the question is whether Frege's first-order calculus is still deductively sufficient as far as the first-order completeness is (...) concerned. In this short note we confirm that the missing axiom is derivable from his stated axioms and inference rules, and hence the logic system in the Begriffsschrift is indeed first-order complete. (shrink)

König, D. [1926. ‘Sur les correspondances multivoques des ensembles’, Fundamenta Mathematica, 8, 114–34] includes a result subsequently called König's Infinity Lemma. Konig, D. [1927. ‘Über eine Schlussweise aus dem Endlichen ins Unendliche’, Acta Litterarum ac Scientiarum, Szeged, 3, 121–30] includes a graph theoretic formulation: an infinite, locally finite and connected graph includes an infinite path. Contemporary applications of the infinity lemma in logic frequently refer to a consequence of the infinity lemma: an infinite, locally finite tree with a root has (...) a infinite branch. This tree lemma can be traced to [Beth, E. W. 1955. ‘Semantic entailment and formal derivability’, Mededelingen der Kon. Ned. Akad. v. Wet., new series 18, 13, 309–42]. It is argued that Beth independently discovered the tree lemma in the early 1950s and that it was later recognized among logicians that Beth's result was a consequence of the infinity lemma. The equivalence of these lemmas is an easy consequence of a well known result in graph theory: every connected, locally finite graph has among its partial subgraphs a spanning tree. (shrink)

Chapin reviewed this 1972 ZEITSCHRIFT paper that proves the completeness theorem for the logic of variable-binding-term operators created by Corcoran and his student John Herring in the 1971 LOGIQUE ET ANALYSE paper in which the theorem was conjectured. This leveraging proof extends completeness of ordinary first-order logic to the extension with vbtos. Newton da Costa independently proved the same theorem about the same time using a Henkin-type proof. This 1972 paper builds on the 1971 “Notes on a Semantic Analysis of (...) Variable Binding Term Operators” (Co-author John Herring), Logique et Analyse 55, 646–57. MR0307874 (46 #6989). A variable binding term operator (vbto) is a non-logical constant, say v, which combines with a variable y and a formula F containing y free to form a term (vy:F) whose free variables are exact ly those of F, excluding y. Kalish-Montague 1964 proposed using vbtos to formalize definite descriptions “the x: x+x=2”, set abstracts {x: F}, minimization in recursive function theory “the least x: x+x>2”, etc. However, they gave no semantics for vbtos. Hatcher 1968 gave a semantics but one that has flaws described in the 1971 paper and admitted by Hatcher. In 1971 we give a correct semantic analysis of vbtos. We also give axioms for using them in deductions. And we conjecture strong completeness for the deductions with respect to the semantics. The conjecture, proved in this paper with Hatcher’s help, was proved independently about the same time by Newton da Costa. (shrink)

We present a novel variant of decision making based on the mathematical theory of separable Hilbert spaces. This mathematical structure captures the effect of superposition of composite prospects, including many incorporated intentions, which allows us to describe a variety of interesting fallacies and anomalies that have been reported to particularize the decision making of real human beings. The theory characterizes entangled decision making, non-commutativity of subsequent decisions, and intention interference. We demonstrate how the violation of the Savage’s sure-thing principle, known (...) as the disjunction effect, can be explained quantitatively as a result of the interference of intentions, when making decisions under uncertainty. The disjunction effects, observed in experiments, are accurately predicted using a theorem on interference alternation that we derive, which connects aversion-to-uncertainty to the appearance of negative interference terms suppressing the probability of actions. The conjunction fallacy is also explained by the presence of the interference terms. A series of experiments are analyzed and shown to be in excellent agreement with a priori evaluation of interference effects. The conjunction fallacy is also shown to be a sufficient condition for the disjunction effect, and novel experiments testing the combined interplay between the two effects are suggested. (shrink)

The Research Assessment Exercise was introduced in 1986 by Thatcher, and was continued by Blair. So it has now been running for 21 years. During this time, the rules governing the RAE have changed considerably, and the interval between successive RAEs has also varied. These changes are not of great importance as far as the argument of this paper is concerned. We will concentrate on the main features of the RAE which can be summarised as follows.

The simplest quantified modal logic combines classical quantification theory with the propositional modal logic K. The models of simple QML relativize predication to possible worlds and treat the quantifier as ranging over a single fixed domain of objects. But this simple QML has features that are objectionable to actualists. By contrast, Kripke-models, with their varying domains and restricted quantifiers, seem to eliminate these features. But in fact, Kripke-models also have features to which actualists object. Though these philosophers have introduced variations (...) on Kripke-models to eliminate their objectionable features, the most well-known variations all have difficulties of their own. The present authors reexamine simple QML and discover that, in addition to having a possibilist interpretation, it has an actualist interpretation as well. By introducing a new sort of existing abstract entity, the contingently nonconcrete, they show that the seeming drawbacks of the simplest QML are not drawbacks at all. Thus, simple QML is independent of certain metaphysical questions. (shrink)

Scientific knowledge develops in an increasingly fragmentary way.A multitude of scientific disciplines branch out. Curiosity for thisdevelopment leads into quests for a unifying understanding. To a certain extent, foundational studies provide such unification. There is a tendency, however, also of a fragmentary growth of foundational studies, like in a multitude of disciplinaryfoundations. We suggest to look at the foundational problem, not primarily as a search for foundations for one discipline in another, as in some reductionist approach, but as a steady (...) revelation ofpresuppositions for individual scientific theories – which are boundto meet, sooner or later, in a common language. A decisive point hereis our holistic conception of language, as a whole of description-interpretation processes which are entangled(complementary) in the language itself. For every language there is alinguistic complementarity. We suggest this unique form ofentanglement as a unifying presupposition, ultimately foundational forall communicable knowledge. Involved is a linguistic realism, in terms ofwhich we critically examine ``language-world'' problems, as exposed byWittgenstein, and Russell, about a foundational interdependence of language andreality (world). Throughout, we attach to the developmentof foundational studies of mathematics, logics, and the naturalsciences. In particular, we study the interpretation problem foraxioms of infinity in some detail. We emphasize that the holistic concept of language contradicts Carnap's semiotic fragmentation thesis (thus, no clean cutbetween syntax, semantics, pragmatics). (shrink)

Mathematicians and physical scientists depend heavily on the formal symbolism of mathematics in order to express and develop their theories. For this and other reasons the last hundred years has seen a growing interest in the nature of formal language and the way it expresses meaning; particularly the objective, shared aspect of meaning as opposed to subjective, personal aspects. This dichotomy suggests the question: do the objective philosophical theories of meaning offer concepts which can be applied in psychological theories of (...) meaning? In recent years cognitive scientists such as Chomsky [1980], Fodor [1981] and MacNamara [1982] have used philosophical approaches to the meaning of formal language expressions as the basis for their psychological theories. Following this lead it seems appropriate to review some of the main treatments of meaning with a view to their transferability. (shrink)

Recently, connectionist models have been developed that seem to exhibit structuresensitive cognitive capacities without executing a program. This paper examines one such model and argues that it does execute a program. The argument proceeds by showing that what is essential to running a program is preserving the functional structure of the program. It has generally been assumed that this can only be done by systems possessing a certain temporalcausal organization. However, counterfactualpreserving functional architecture can be instantiated in other ways, for (...) example geometrically, which are realizable by connectionist networks. (shrink)

An axiomatic theory of abstract objects is developed and used to construct models of Plato's Forms, Leibniz's Monads, Possible Worlds, Frege's Senses, stories, and fictional characters. The theory takes six primitive metaphysical notions: object ; n-place relations ,G,...); x,...x exemplify F x...x); x exists ; it is necessary that "); and x encodes F "). Properties and propositions are one place and zero place relations, respectively.objects are objects which necessarily fail to exist E!x"). The two most important proper axioms are (...) that no possibly existing object encodes any properties E!x F)xF)), and for every expressible condition on properties, there is an abstract object which encodes just the properties satisfying the condition x) )), where has no free x's). Semantically, an abstract object encodes a property iff the property is an element of the set of properties correlated with the object. Two abstract objects will be identical just in case they encode the same properties. Abstract objects may exemplify properties as well--being non-red, not having spatial location, etc. ;The models of Forms, Monads, and Possible Worlds consist in: identifying these philosophical entities as different species of abstract objects through definitions of the object language, and either showing that there are consequences of the theory which are reasonable facsimiles of assertions made by important philosophers or showing that there are many interesting theorems which capture our own intuitions about these entities . The model for Frege's Senses consists in: developing a language in which the abstract objects generated by the theory serve as the senses of names and descriptions, and showing that the data may be consistently translated into the language preserving truth. Finally, the model for fictional characters consists in: assigning abstract objects to serve as the denotations of names of stories and characters, and showing that such an assignment allows us to understand how the intuitively true sentences about stories and characters have the truth value that they do. ;The material is structured as follows. In Chapters I, III, and V, the elementary, modal, and typed theories of abstract objects are developed, respectively. In Chapters II, IV, and VI, these theories are applied, respectively. Each of the theoretical chapters is subdivided as follows: 1--The Language; 2--The Semantics; 3--The Logic; and 4--The Proper Axioms. Thus, the proper axioms of the theory are couched in a precisely defined and interpreted language, and the logic allow us to prove the consequences of the theory. (shrink)

The study is focused on the relation between theology and mathematics in the situation of increasing secularization. My main concern in the second part of this paper is the early-twentieth-century foundational crisis of mathematics. The hypothesis that pure mathematics partially fulfilled the functions of theology at that time is tested on the views of the leading figures of the three main foundationalist programs: Russell, Hilbert and Brouwer.

This paper provides a logic framework for investigations of game theoretical problems. We adopt an infinitary extension of classical predicate logic as the base logic of the framework. The reason for an infinitary extension is to express the common knowledge concept explicitly. Depending upon the choice of axioms on the knowledge operators, there is a hierarchy of logics. The limit case is an infinitary predicate extension of modal propositional logic KD4, and is of special interest in applications. In Part I, (...) we develop the basic framework, and show some applications: an epistemic axiomatization of Nash equilibrium and formal undecidability on the playability of a game. To show the formal undecidability, we use a term existence theorem, which will be proved in Part II. (shrink)

Scientific inquiry is represented as a process of rational hypothesis revision in the face of data. For the concept of rationality, we rely on the theory of belief dynamics as developed in [5, 9]. Among other things, it is shown that if belief states are left unclosed under deductive logic then scientific theories can be expanded in a uniform, consistent fashion that allows inquiry to proceed by any method of hypothesis revision based on "kernel" contraction. In contrast, if belief states (...) are closed under logic, then no such expansion is possible. (shrink)

Let M be a first-order structure; we denote by DEF(M) the set of all first-order definable relations and functions within M. Let π be any one-to-one function from N into the set of prime integers. Let ∣ and $\bullet$ be respectively the divisibility relation and multiplication as function. We show that the sets DEF(N,π,∣) and $\mathrm{DEF}(\mathbb{N},\pi,\bullet)$ are equal. However there exists function π such that the set DEF(N,π,∣), or, equivalently, $\mathrm{DEF}(\mathbb{N},\pi,\bullet)$ is not equal to $\mathrm{DEF}(\mathbb{N},+,\bullet)$ . Nevertheless, in all cases (...) there is an $\{\pi,\bullet\}$ -definable and hence also {π,∣}-definable structure over N which is isomorphic to $\langle\mathbb{N},+,\bullet\rangle$ . Hence theories TH(N,π,∣) and $\mathrm{TH}(\mathbb{N},\pi,\bullet)$ are undecidable. The binary relation of equipotence between two positive integers saying that they have equal number of prime divisors is not definable within the divisibility lattice over positive integers. We prove it first by comparing the lower bound of the computational complexity of the additive theory of positive integers and of the upper bound of the computational complexity of the theory of the mentioned lattice. The last section provides a self-contained alternative proof of this latter result based on a decision method linked to an elimination of quantifiers via specific tables. (shrink)

A one-premiss rule is said to be archetypal for a consequence relation when not only is the conclusion of any application of the rule a consequence of the premiss, but whenever one formula has another as a consequence, these formulas are respectively equivalent to a premiss and a conclusion of some application of the rule. We are concerned here with the consequence relation of classical propositional logic and with the task of extending the above notion of archetypality to rules with (...) more than one premiss, and providing an informative characterization of the set of rules falling under the more general notion. (shrink)